Here is a short lesson I created for church organists who on occasion may be desirous to employ a simple
re-harmonization device without getting too carried away. I apologize for the little rant in the beginning of the video about organists getting in the way of the congregation. I too am a frequent offender of this practice. It is part of the organist ego. The devil makes us do it. Bach's congregation had the same plaint.
This device is simple. Whenever the melody is on the third tone of a chord, or you change the chord to make that note the third, leave the melody where it is but raise the chord up half a step to it's minor equivalent, then drop it down to its dominant seventh. Keep the voicing open as that will leave a lot of room for inner linear movement and a lot of room other chordal substitutions and leading. If you don't know what that means, that is okay, listen to your ear. It knows.
I often throw something like this in toward the the end of a verse to signal to the congregation that I am about to do something such as a key change or interlude. I usually only throw in interludes when the liturgical movement calls for it because the people on the dais need more time to get where they're going or to do what they're doing. If a choir is processing and they just hit the stairs to the balcony, I may do the same thing.
Personally, if I am pew-side of a church, I like to sing the harmony to the hymns and when the organist doesn't play what's on the page it renders me mute. Organists need to be cognizant of the text, too. I remember being at a music convention for Pastoral Musicians and on the fifth verse of a hymn, the text stated something about not toiling or mourning for, the gentle presence of God will carry you through the tough times. I thought it ironic that the organist was re-harmonizing, ratcheting up the crescendo pedal and tossing out trite-trumpet-triplets all as we were singing words such as "quiet" and "gentle."
Read your texts, love your people, help them to sing, hold their hand if necessary. Think of the church in the theater model; the congregants are the actors, God is the audience and you are the prompter. Prompt, don't hijack.